Proving that a subgroup is conjugate-dense

This article discusses general strategies for proving that a subgroup of a group is a conjugate-dense subgroup, i.e., that every element of the whole group is conjugate to some element of the subgroup.

Note that for a finite group, no proper subgroup can be conjugate-dense. More generally, in any group, no proper subgroup of finite index can be conjugate-dense. Further information: Union of all conjugates is proper

Most of the strategies discussed here work not just for subgroups, but for arbitrary subsets. In other words, given a group and a subset , these strategies help prove that every element of is conjugate to some element of . While the most special case is that where is a subgroup of , other cases of interest arise, for example, when is a union of a few well-chosen subgroups.

Related techniques

The general strategy

Suppose is a group and is a subgroup (or more generally, subset) of . Then, acts on itself by conjugation. We want to show that every element of is in the orbit of some element of . Equivalently, we want to show that starting with any element , we can find an element such that .

The step-by-step approach

In this approach, we think of the elements of as extreme elements, and create a gradation in the elements of . Next, we show that, starting with any arbitrary element ,